scholarly journals DenseQ-subalgebras of Banach andC*-algebras and unbounded derivations of Banach andC*-algebras

1993 ◽  
Vol 36 (2) ◽  
pp. 261-276 ◽  
Author(s):  
E. Kissin ◽  
V. S. Shulman
Keyword(s):  
Banach Algebra ◽  
Unital Algebra ◽  
Injective Homomorphism ◽  
Finite Dimensional ◽  
Dense Subalgebra ◽  
Unbounded Derivations ◽  
One To One

The paper studies denseQ-subalgebras of Banach andC*-algebras. It proves that the domainD(δ) of a closed unbounded derivation δ of a Banach unital algebraAautomatically contains the identity and is aQ-subalgebra ofA, so thatSpA(x) =SpD(δ)(x) for allx∈D(δ). The paper shows that every finite-dimensional semisimple representation of aQ-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normalQ*-subalgebraBof aC*-algebra, then ‖x‖≦‖π(x)‖ for allx∈B. The paper studies the link between closed ideals of a Banach algebraAand of its dense subalgebraB. In particular, ifAis aC*-algebra andBis a locally normal *-subalgebra ofA, thenI→I∩Bis a one-to-one mapping of the set of all closed two-sided ideals inAonto the set of all closed two-sided ideals inBand.

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (1) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

scholarly journals On a problem of Barnes and Duncan

1991 ◽  
Vol 34 (2) ◽  
pp. 321-323
Author(s):  
R. G. McLean
Keyword(s):  
Banach Algebra ◽  
Free Monoid ◽  
Finite Dimensional ◽  
Image Position ◽  
Funct Anal

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach *-algebra ℓ1(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Funct. Anal.18 (1975), 96–113.) dealing with the case where P has two elements.

scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

scholarly journals Completion of normed algebras of polunomials

1975 ◽  
Vol 20 (4) ◽  
pp. 504-510 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Complex Field ◽  
Linear Functionals ◽  
Bounded Linear ◽  
One To One ◽  
Formal Power

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

scholarly journals The Rank+Nullity Theorem

2007 ◽  
Vol 15 (3) ◽  
pp. 137-142 ◽  
Author(s):  
Jesse Alama
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
One To One

The Rank+Nullity Theorem The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.

scholarly journals Similarité entre l'algèbre de Volterra et un quotient d'algèbre uniforme

1991 ◽  
Vol 34 (3) ◽  
pp. 383-391 ◽  
Author(s):  
Konin Koua
Keyword(s):  
Banach Algebra ◽  
Half Plane ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Compact Mapping ◽  
Dense Ideal ◽  
Right Hand ◽  
Commutative Banach Algebras ◽  
One To One

Two commutative Banach algebras A and B are said to be similar if there exists a Banach algebra D such that [xD]− = D for some x in D, and two one-to-one continuous homomorphisms φ:D→A and ψ:D→B such that φ(D) is a dense ideal of A and ψ(D) a dense ideal of B.We prove in this paper that the Volterra algebra is similar to A0/e-z A0 where A0 is the commutative uniform, separable Banach algebra of all continuous functions on the closed right-hand half plane , analytic on H and vanishing at infinity. We deduce from this result that multiplication by an element of A0/e-z A0 is a compact mapping.

scholarly journals Duality for automorphisms on a compact C*-dynamical system

1988 ◽  
Vol 8 (2) ◽  
pp. 173-189 ◽  
Author(s):  
David E. Evans ◽  
Akitaka Kishimoto
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Positive Element ◽  
Constant Multiple ◽  
Unital Algebra ◽  
Finite Dimensional ◽  
Extended Action ◽  
Image Position ◽  
Fixed Point Algebra ◽  
Spectral Subspaces

When considering an action α of a compact group G on a C*-algebra A, the notion of an α-invariant Hilbert space in A has proved extremely useful [1, 4, 8, 14, 17, 18]. Following Roberts [13] a Hilbert space in (a unital algebra) A is a closed subspace H of A such that x*y is a scalar for all x, y in H. For example if G is abelian, and α is ergodic in the sense that the fixed point algebra Aα is trivial, then A is generated as a Banach space by a unitary in each of the spectral subspaceswhich are then invariant one dimensional Hilbert spaces. If G is not abelian, then Hilbert spaces (which are always assumed to be invariant) do not necessarily exist, even for ergodic actions. For non-ergodic actions, it is also desirable to relax the requirement to x*y being a constant multiple of some positive element of Aα. More generally, if γ is a finite dimensional matrix representation of G and n is a positive integer, we define to be the subspacewhere d is the dimension d(γ) of γ, and Mnd denotes n×d complex matrices. (Usually we will denote the extended action of αg to αg ⊗ 1 on A⊗Mnd again by αg.) Let .

scholarly journals On one-sided primitivity of Banach algebras

2010 ◽  
Vol 53 (1) ◽  
pp. 111-123 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Semigroup Algebra ◽  
Alternative Proof ◽  
Infinite Dimensional ◽  
Finite Dimensional

AbstractLet S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

scholarly journals A note on annihilator Banach algebras

1988 ◽  
Vol 38 (1) ◽  
pp. 77-81
Author(s):  
Pak-Ken Wong
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dense Subalgebra ◽  
Semisimple Banach Algebra

LetAbe a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebraBwith | · | such that ‖ · ‖ majorises | · | onA. The purpose of this paper is to investigate the annihilator property between the algebrasAandB.

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